Integration by Parts

Integration by parts is a technique used to find the integral of a product of functions. It is based on the product rule of differentiation and is particularly useful when direct integration is difficult or impossible.

The Formula

The integration by parts formula is derived from the product rule for differentiation:

$$\int u(x) \cdot v'(x) \, dx = u(x) \cdot v(x) - \int v(x) \cdot u'(x) \, dx$$

Where:

$u(x)$ and $v'(x)$ are functions of $x$
$u'(x)$ is the derivative of $u(x)$
$v(x)$ is the antiderivative of $v'(x)$

This technique transforms the original integral into another one that may be easier to solve, often used when integrating products like $x \cdot e^x$, $x \cdot \sin(x)$, or $\ln(x) \cdot x$.

When to Use Integration by Parts

Integration by parts is particularly useful for integrals that involve:

Products where one function gets simpler when differentiated (for u)
Products where one function has a known antiderivative (for v')

Common Patterns

The LIATE rule can help you choose which function to assign as u:

Logarithmic functions: ln(x), log₁₀(x)
Inverse trigonometric functions: arcsin(x), arctan(x)
Algebraic functions: x, x², polynomial functions
Trigonometric functions: sin(x), cos(x), tan(x)
Exponential functions: e^x, a^x

Functions earlier in the list are generally better choices for u, as the derivative tends to simplify the function.

Multiple Applications

Sometimes, a single application of integration by parts doesn't fully solve the integral. You may need to apply the technique multiple times.

Example: $\int x^2 e^x \, dx$

First application of integration by parts (let u = x² and v' = e^x):

$$\int x^2 e^x \, dx = x^2 e^x - \int 2x \cdot e^x \, dx$$

For the new integral $\int 2x \cdot e^x \, dx$, apply integration by parts again (let u = 2x and v' = e^x):

$$\int 2x \cdot e^x \, dx = 2x \cdot e^x - \int 2 \cdot e^x \, dx = 2x \cdot e^x - 2e^x$$

Combining the results:

$$\int x^2 e^x \, dx = x^2 e^x - 2x \cdot e^x + 2e^x + C = e^x(x^2 - 2x + 2) + C$$

Notes and Resources

Key Takeaways

Integration by parts is a powerful technique for integrating products of functions
The formula is derived from the product rule of differentiation
Use the LIATE rule to help choose which function to assign as u
Sometimes multiple applications of the technique are needed
This technique is essential for solving many integrals in calculus